We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime: \begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}a_{n-2},\\ \vdots\\ a_{n}a_{n-1}a_{n-2} \cdots a_{1}.\end{eqnarray*}
According to user David W. Wilson of the OEIS, there exist only $83$ right-truncatable prime numbers: Mr. D. W. Wilson also claims that the largest such prime is $73 \, 939 \, 133$. Since the first of these two claims provides a clear-cut answer to a problem which I had come up with some time ago (before I even knew these primes were called right-truncatable; in point of fact, I had christened 'em as totally prime (go figure!)), I would like to know if you know of (or can supply) a non-computer-assisted straightforward proof of the finiteness of the set of right-truncatable prime numbers...
It may be mentioned that, in order to test those two assertions of Mr. D. W. Wilson, I wrote a Mathematica program that determines whether or not a given prime number is right-truncatable and had it look for primes of that type in the interval $[10^{8}, 10^{9})$:
For[k = 100000000, k < 1000000000, k++,
If[PrimeQ[k],
aux = 1;
flag = 2;
j = k;
While[flag <= IntegerLength[j], h = (j - Mod[j, 10])/10;
If[PrimeQ[h], aux = aux + 1; j = h; flag = flag - 1, flag = 1000]];
If[aux == IntegerLength[k], Print[k]];
]]
As any die-hard fan of the OEIS would have expected, the above program found no right-truncatable prime number in that interval!
Please, let me thank you in advance for your knowledgeable replies.